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A long cylinder of radius R_1 is displaced along its axis with a constant velocity v_0 inside a stationary co-axial cylinder of radius R_2. The space between the cylinders is filled with viscous liquid. Find the velocity of the liquid as a function of the distance r from the axis of the cylinders. The flow is laminar.

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Answer Text

Solution :
Let us consider a coaxial cylinder of radius r and thickness `dr`, then force of friction or viscous force on this elemental layer, `F=2pi r l eta(dv)/(dr)`. <br> This force must be constant from layer to layer so that steady motion may be possible. <br> or, `(Fdr)/(r)=2pi l eta dv` (1) <br> Integrating, <br> `Funderset(R_2)overset(r)int(dr)/(r)=2pi l eta underset(0)overset(v)int dr` <br> or, `F1n((r)/(R_2))=2pi l eta v` (2) <br> Putting `r=R_1`, we get <br> `F1n(R_1)/(R_2)=2pi l eta v_0` <br> From (2) by (3) we get, <br> `v=v_0(1nr//R_2)/(1nR_1//R_2)` <br> Note: The force F is supplied by the agency which tries to carry the inner cylinder with velocity `v_0`. <br> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/IROD_V01_C01_E01_332_S01.png" width="80%">

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